5.12/2.18 YES 5.46/2.20 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.46/2.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.46/2.20 5.46/2.20 5.46/2.20 Left Termination of the query pattern 5.46/2.20 5.46/2.20 perm(g,a) 5.46/2.20 5.46/2.20 w.r.t. the given Prolog program could successfully be proven: 5.46/2.20 5.46/2.20 (0) Prolog 5.46/2.20 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.46/2.20 (2) PiTRS 5.46/2.20 (3) DependencyPairsProof [EQUIVALENT, 1 ms] 5.46/2.20 (4) PiDP 5.46/2.20 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.46/2.20 (6) AND 5.46/2.20 (7) PiDP 5.46/2.20 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.46/2.20 (9) PiDP 5.46/2.20 (10) PiDPToQDPProof [SOUND, 0 ms] 5.46/2.20 (11) QDP 5.46/2.20 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.46/2.20 (13) YES 5.46/2.20 (14) PiDP 5.46/2.20 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.46/2.20 (16) PiDP 5.46/2.20 (17) PiDPToQDPProof [SOUND, 0 ms] 5.46/2.20 (18) QDP 5.46/2.20 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.46/2.20 (20) YES 5.46/2.20 (21) PiDP 5.46/2.20 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.46/2.20 (23) PiDP 5.46/2.20 (24) PiDPToQDPProof [SOUND, 0 ms] 5.46/2.20 (25) QDP 5.46/2.20 (26) MRRProof [EQUIVALENT, 0 ms] 5.46/2.20 (27) QDP 5.46/2.20 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 5.46/2.20 (29) TRUE 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (0) 5.46/2.20 Obligation: 5.46/2.20 Clauses: 5.46/2.20 5.46/2.20 perm([], []). 5.46/2.20 perm(L, .(H, T)) :- ','(append2(V, .(H, U), L), ','(append1(V, U, W), perm(W, T))). 5.46/2.20 append1([], L, L). 5.46/2.20 append1(.(H, L1), L2, .(H, L3)) :- append1(L1, L2, L3). 5.46/2.20 append2([], L, L). 5.46/2.20 append2(.(H, L1), L2, .(H, L3)) :- append2(L1, L2, L3). 5.46/2.20 5.46/2.20 5.46/2.20 Query: perm(g,a) 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (1) PrologToPiTRSProof (SOUND) 5.46/2.20 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.46/2.20 5.46/2.20 perm_in_2: (b,f) 5.46/2.20 5.46/2.20 append2_in_3: (f,f,b) 5.46/2.20 5.46/2.20 append1_in_3: (b,b,f) 5.46/2.20 5.46/2.20 Transforming Prolog into the following Term Rewriting System: 5.46/2.20 5.46/2.20 Pi-finite rewrite system: 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 5.46/2.20 5.46/2.20 5.46/2.20 5.46/2.20 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.46/2.20 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (2) 5.46/2.20 Obligation: 5.46/2.20 Pi-finite rewrite system: 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (3) DependencyPairsProof (EQUIVALENT) 5.46/2.20 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AAG(V, .(H, U), L) 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> APPEND1_IN_GGA(V, U, W) 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> U3_GA(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.46/2.20 5.46/2.20 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 5.46/2.20 5.46/2.20 U5_AAG(x1, x2, x3, x4, x5) = U5_AAG(x5) 5.46/2.20 5.46/2.20 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 5.46/2.20 5.46/2.20 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 5.46/2.20 5.46/2.20 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (4) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AAG(V, .(H, U), L) 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> APPEND1_IN_GGA(V, U, W) 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> U3_GA(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.46/2.20 5.46/2.20 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 5.46/2.20 5.46/2.20 U5_AAG(x1, x2, x3, x4, x5) = U5_AAG(x5) 5.46/2.20 5.46/2.20 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 5.46/2.20 5.46/2.20 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 5.46/2.20 5.46/2.20 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (5) DependencyGraphProof (EQUIVALENT) 5.46/2.20 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (6) 5.46/2.20 Complex Obligation (AND) 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (7) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (8) UsableRulesProof (EQUIVALENT) 5.46/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (9) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) 5.46/2.20 5.46/2.20 R is empty. 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (10) PiDPToQDPProof (SOUND) 5.46/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (11) 5.46/2.20 Obligation: 5.46/2.20 Q DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND1_IN_GGA(.(L1), L2) -> APPEND1_IN_GGA(L1, L2) 5.46/2.20 5.46/2.20 R is empty. 5.46/2.20 Q is empty. 5.46/2.20 We have to consider all (P,Q,R)-chains. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (12) QDPSizeChangeProof (EQUIVALENT) 5.46/2.20 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 5.46/2.20 5.46/2.20 From the DPs we obtained the following set of size-change graphs: 5.46/2.20 *APPEND1_IN_GGA(.(L1), L2) -> APPEND1_IN_GGA(L1, L2) 5.46/2.20 The graph contains the following edges 1 > 1, 2 >= 2 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (13) 5.46/2.20 YES 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (14) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (15) UsableRulesProof (EQUIVALENT) 5.46/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (16) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) 5.46/2.20 5.46/2.20 R is empty. 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (17) PiDPToQDPProof (SOUND) 5.46/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (18) 5.46/2.20 Obligation: 5.46/2.20 Q DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 APPEND2_IN_AAG(.(L3)) -> APPEND2_IN_AAG(L3) 5.46/2.20 5.46/2.20 R is empty. 5.46/2.20 Q is empty. 5.46/2.20 We have to consider all (P,Q,R)-chains. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (19) QDPSizeChangeProof (EQUIVALENT) 5.46/2.20 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 5.46/2.20 5.46/2.20 From the DPs we obtained the following set of size-change graphs: 5.46/2.20 *APPEND2_IN_AAG(.(L3)) -> APPEND2_IN_AAG(L3) 5.46/2.20 The graph contains the following edges 1 > 1 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (20) 5.46/2.20 YES 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (21) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 perm_in_ga([], []) -> perm_out_ga([], []) 5.46/2.20 perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) 5.46/2.20 U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.46/2.20 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.46/2.20 5.46/2.20 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.46/2.20 5.46/2.20 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.46/2.20 5.46/2.20 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.46/2.20 5.46/2.20 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (22) UsableRulesProof (EQUIVALENT) 5.46/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (23) 5.46/2.20 Obligation: 5.46/2.20 Pi DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) 5.46/2.20 U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) 5.46/2.20 PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 append1_in_gga([], L, L) -> append1_out_gga([], L, L) 5.46/2.20 append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) 5.46/2.20 append2_in_aag([], L, L) -> append2_out_aag([], L, L) 5.46/2.20 append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) 5.46/2.20 U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) 5.46/2.20 U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) 5.46/2.20 5.46/2.20 The argument filtering Pi contains the following mapping: 5.46/2.20 [] = [] 5.46/2.20 5.46/2.20 append2_in_aag(x1, x2, x3) = append2_in_aag(x3) 5.46/2.20 5.46/2.20 .(x1, x2) = .(x2) 5.46/2.20 5.46/2.20 append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) 5.46/2.20 5.46/2.20 U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) 5.46/2.20 5.46/2.20 append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) 5.46/2.20 5.46/2.20 append1_out_gga(x1, x2, x3) = append1_out_gga(x3) 5.46/2.20 5.46/2.20 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.46/2.20 5.46/2.20 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.46/2.20 5.46/2.20 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.46/2.20 5.46/2.20 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.46/2.20 5.46/2.20 5.46/2.20 We have to consider all (P,R,Pi)-chains 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (24) PiDPToQDPProof (SOUND) 5.46/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (25) 5.46/2.20 Obligation: 5.46/2.20 Q DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 U1_GA(append2_out_aag(V, .(U))) -> U2_GA(append1_in_gga(V, U)) 5.46/2.20 U2_GA(append1_out_gga(W)) -> PERM_IN_GA(W) 5.46/2.20 PERM_IN_GA(L) -> U1_GA(append2_in_aag(L)) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 append1_in_gga([], L) -> append1_out_gga(L) 5.46/2.20 append1_in_gga(.(L1), L2) -> U4_gga(append1_in_gga(L1, L2)) 5.46/2.20 append2_in_aag(L) -> append2_out_aag([], L) 5.46/2.20 append2_in_aag(.(L3)) -> U5_aag(append2_in_aag(L3)) 5.46/2.20 U4_gga(append1_out_gga(L3)) -> append1_out_gga(.(L3)) 5.46/2.20 U5_aag(append2_out_aag(L1, L2)) -> append2_out_aag(.(L1), L2) 5.46/2.20 5.46/2.20 The set Q consists of the following terms: 5.46/2.20 5.46/2.20 append1_in_gga(x0, x1) 5.46/2.20 append2_in_aag(x0) 5.46/2.20 U4_gga(x0) 5.46/2.20 U5_aag(x0) 5.46/2.20 5.46/2.20 We have to consider all (P,Q,R)-chains. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (26) MRRProof (EQUIVALENT) 5.46/2.20 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 5.46/2.20 5.46/2.20 Strictly oriented dependency pairs: 5.46/2.20 5.46/2.20 U1_GA(append2_out_aag(V, .(U))) -> U2_GA(append1_in_gga(V, U)) 5.46/2.20 U2_GA(append1_out_gga(W)) -> PERM_IN_GA(W) 5.46/2.20 5.46/2.20 5.46/2.20 Used ordering: Polynomial interpretation [POLO]: 5.46/2.20 5.46/2.20 POL(.(x_1)) = 2 + x_1 5.46/2.20 POL(PERM_IN_GA(x_1)) = 2*x_1 5.46/2.20 POL(U1_GA(x_1)) = 2*x_1 5.46/2.20 POL(U2_GA(x_1)) = 1 + 2*x_1 5.46/2.20 POL(U4_gga(x_1)) = 2 + x_1 5.46/2.20 POL(U5_aag(x_1)) = 2 + x_1 5.46/2.20 POL([]) = 0 5.46/2.20 POL(append1_in_gga(x_1, x_2)) = x_1 + x_2 5.46/2.20 POL(append1_out_gga(x_1)) = x_1 5.46/2.20 POL(append2_in_aag(x_1)) = x_1 5.46/2.20 POL(append2_out_aag(x_1, x_2)) = x_1 + x_2 5.46/2.20 5.46/2.20 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (27) 5.46/2.20 Obligation: 5.46/2.20 Q DP problem: 5.46/2.20 The TRS P consists of the following rules: 5.46/2.20 5.46/2.20 PERM_IN_GA(L) -> U1_GA(append2_in_aag(L)) 5.46/2.20 5.46/2.20 The TRS R consists of the following rules: 5.46/2.20 5.46/2.20 append1_in_gga([], L) -> append1_out_gga(L) 5.46/2.20 append1_in_gga(.(L1), L2) -> U4_gga(append1_in_gga(L1, L2)) 5.46/2.20 append2_in_aag(L) -> append2_out_aag([], L) 5.46/2.20 append2_in_aag(.(L3)) -> U5_aag(append2_in_aag(L3)) 5.46/2.20 U4_gga(append1_out_gga(L3)) -> append1_out_gga(.(L3)) 5.46/2.20 U5_aag(append2_out_aag(L1, L2)) -> append2_out_aag(.(L1), L2) 5.46/2.20 5.46/2.20 The set Q consists of the following terms: 5.46/2.20 5.46/2.20 append1_in_gga(x0, x1) 5.46/2.20 append2_in_aag(x0) 5.46/2.20 U4_gga(x0) 5.46/2.20 U5_aag(x0) 5.46/2.20 5.46/2.20 We have to consider all (P,Q,R)-chains. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (28) DependencyGraphProof (EQUIVALENT) 5.46/2.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.46/2.20 ---------------------------------------- 5.46/2.20 5.46/2.20 (29) 5.46/2.20 TRUE 5.46/2.22 EOF